Is #f(x) = (x-3)^3 - 6x^3# concave or convex at #x=-1#?
1 Answer
Feb 24, 2016
convex at x = -1
Explanation:
To test if a function is concave / convex at f(a) , require to find the value of f''(a).
• If f''(a) > 0 then f(x) is convex at x = a
• If f''(a) < 0 then f(x) is concave at x = a
hence f(x)
#= (x-3)^3 - 6x^3# f'(x) =
#3(x-3)^2 d/dx(x-3) - 18x^2 = 3(x-3)^2 - 18x^2# and f''(x) = 6(x-3).1 - 36x
hence f''(-1) = 6(-4) - 36(-1) = -24 + 36 = 12
since f''(-1) > 0 then f(x) is convex at x = -1
